Stochastic Programming

605 Pages·1995·13.061 MB·English

by  András Prékopa (auth.)

Stochastic Programming
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 324
Stochastic Programming
by
Andras Prekopa
RUTCOR, Rutgers Center/or Operations Research, Rutgers University,
New Brunswick, NJ, U.S.A.
and
Department o/Operations Research, £Orand Eiitviis University,
Budapest, Hungary
..
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4552-2 ISBN 978-94-017-3087-7 (eBook)
DOI 10.1007/978-94-017-3087-7
Ali Rights Reserved
© 1995 Andrâs Prekopa
Originally published by Kluwer Academic Publishers in 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
inc1uding photocopying, recording or by any information storage and
retrieval system, without written permis sion from the copyright owner.
To my wife Kinga
Preface
At the beginning of the nineteen fifties, when I was a young graduate student in
probability theory and statistics at the Institute for Applied Mathematics of the
Hungarian Academy of Sciences, I was required to work on theoretical and applied
problems, just like the other members of the Institute. Applications which were based
on advanced theory and/or gave rise to novel theoretical developments, were the
most wanted and appreciated. However, our possibilities to satisfy the requirements
imposed on us were strongly limited, because the country was in ruins due to World
War II and the subsequent economic and political changes.
Some of the engineering problems still looked hopeful and I began to deal with
probabilistic and statistical type water resources problems. The chief water engineer
of the city of Budapest was furious when I offered my applied mathematics for the
benefit of the water supply of the city, because the main problem was to secure funds
to build new wells at the banks of the Duna River to the north of Budapest. Another
water engineer was more cooperative but his problem did not attract my interest.
I found a reasonable probabilistic problem in determining the size of a water
reservoir, intended to serve a new thermal power station, in the northern part of the
country. I found an "ad hoc" solution to the problem: what is the smallest size of
the reservoir still enough to deliver the necessary amount of water by a prescribed
large probability? I could have been one of the inventors of the so-called "Moran dam
model", because first I tried to solve the problem along that line, in a theoretically
nice way. But my collegues and I rejected it because no river in Hungary had random
water inputs which could be regarded as a sequence of Li.d. random variables, in the
subsequent periods, let alone the fact that the water flow is a continuous rather than
a discrete process. The lack of applicability of higher level mathematics to the given
problem was disappointing.
In the early months of nineteen fifty-seven I encountered linear programming and
read more about operations research. I was fascinated by seeing that we were able to
viii Preface
find extrema of linear functions over completely arbitrary convex polyhedra. There
were limitations regarding the number of inequalities and variables involved, due
to computational difficulties, but there were none regarding the type of the convex
polyhedra involved. That is what we need in probability and statistics, I told myself,
to be able to solve problems under fairly general restrictions regarding the type of
the random variables involved. There were other colleagues too who had the same in
mind.
Already in the nineteen fifties mathematical programming underwent a rapid de
velopment, and answers have been proposed for the question: what to do with a linear
or nonlinear programming problem when some of its parameters are random variables.
Stochastic programming came to existence. Models and methods have been created
which offered some solutions to the above-mentioned problems too. However, many
years had to pass until stochastic programming became sufficiently well-developed
to satisfy the above-mentioned expectations. How can we define this science? Two
defini tions are presented below.
First definition: stochastic programming is the science which offers solutions for
problems formulated in connection with stochastic systems, where the resulting nu
merical problem to be solved is a mathematical programming problem of nontrivial
size.
Second definition: stochastic programming handles mathematical programming
problems, where some of the parameters are random variables; either we study the
statistical properties of the random optimum value or other random variables that
come up with the problem, or we reformulate it into a decision type problem by taking
into account the joint probability distribution of the random parameters.
The resulting problems in both definitions are called stochastic programming
problems. The mathematical programming problems which serve as starting prob
lems, mentioned in the second definition, are called base problems or underlying
problems.
We suggest that both definitions should be accepted. Disregarding the distribu
tion problem$, which are only covered by the second definition, all other stochastic
programming problems are covered by the first definition.
In view of the first definition, moment problems and probability approximation
schemes are parts of stochastic programming problems as long as mathematical pro
gramming methods are used in them. At the same time these provide us with exam
ples, where there are no underlying deterministic problems because the moments of
a constant random variable are uninteresting and in this case the moment problems
become trivial.
Stochastic programming decision models can be subdivided into static and dy
namic models, and both types may use probabilistic constraints and/or penalties in
the objective function. A probabilistic constraint prescribes by a large probability
that some constraints which are random while the decision is made, should hold by
the time when the random variables in them are realized. Penalties added to the
system costs in the objective function express costs incurred by the violations of the
constraints when the random variables are realized.
Preface ix
The use of probabilistic constraint versus penalties is frequently debated. Critics
say about models where only probabilistic constraints are used that: (a) the prob
ability level in the probabilistic constraint is arbitrarily chosen, hence the problem is
not well defined; (b) since we usually prescribe that some relations should hold by
large probability, the tail of a probability distribution plays an important role in the
decision making; however, the tail probabilities are frequently hard to estimate with
satisfactory precision; (c) only the system costs are taken into account and the costs
incurred by the uncertainty are left out of consideration.
Models using only penalties in the objective function, without probabilistic con
straint, can also be criticized: (0) the penalty is an expectation of some random
variable which measures how m'llch the constraints are unsatisfied; the expectation is
a long term average, hence the model is realistic only if the system operates for a long
period of time (note that if a probabilistic constraint is used with a large probability
level, then the event that has at least the prescribed probability can be regarded a
practically sure event; thus, the event on which the probabilistic constraint is im
posed is practically sure and even if we operate the system in a single period, our
decision principle provides us with a reasonable result); ((3) the cost of violation of
some constraints in the underlying problem is frequently unknown; (,) no reliability
requirement is imposed on the system, hence it may fail (e.g., an insurance compa
ny may go bankrupt) incurring such large costs which are not represented by the
penalties.
The above criticisms are sometimes justified but not always. The following points
can be made: (a) there are cases in which the optimum values and the optimal
solutions are not significantly sensitive for the choice of a large enough probability
level in the probabilistic constraint; on the other hand, sometimes we can tell how
rarely we may allow some of the stochastic constraints to be violated so that the
system should survive; (b) sometimes the tail probabilities can be well approximated,
e.g., by the use of moments which in turn, can frequently be well estimated; (c) the
cost of uncertainty may be unknown; (0) sometimes we can operate the system for a
long period of time; ((3) there are cases where the penalties of constraint violations
are known; (,) there are cases where system reliability can be left out of consideration
because if a constraint is violated, then we can use a compensation for cost.
The above reasoning shows that the best model construction is the one which
combines the use of a probabilistic constraint and penalties in the objective function.
In practice, however, some other aspect, for example the numerical solvability of the
problem also comes into account, and there is a tradeoff between correct statistical
modelling and computability.
The arsenal of practical problems where stochastic programming can be used is
very large and it is difficult to place one model construction over another as more
important.
The book is a graduate textbook. The field is very large, hence a selection ofthe
topics was unavoidable.
An introduction to linear programming is presented in the first three chapters.
My intention was to make it easily accessible for probabilists and statisticians and
attract their interest to work on stochastic programming. The field needs them very
x Preface
much. In addition to linear programming, there are other prerequisites to read this
book: nonlinear programming, probability theory, and statistics. Due to limitations
regarding the size of the book I declined to include further introductory chapters.
Chapter 4 is devoted to logconcave probability measures and their extensions.
Multivariate logconcave measures first arose in stochastic programming more than
twenty years ago, and became widely used and investigated in other fields such as
probability theory, statistics, geometry, and mathematical analysis. Logconcave mea
sures and their extensions provide us with the mathematical tools to establish the
convexity of some of the stochastic programming problems, especially probabilistic
constrained problems. Therefore, an in-depth presentation of the relevant most im
portant theorems is appropriate.
Chapter 5 presents fundamentals of optimization type moment problems which are
stochastic programming problems in their own right and are used in other stochastic
programming model constructions. A large part of this chapter is devoted to discrete
moment problems which came to prominence concerning bounding probabilities of
Boolean functions of events and sets in higher dimensional spaces.
Chapter 6 pays more attention to these bounding procedures and presents prob
ability approximation schemes, combining numerical integration and simulation.
Chapter 7 presents statistical decision theoretical principles in a historical frame
work. It explains where the model constructions of stochastic programming come
from.
Chapter 8 summarizes the basic static type stochastic programming model con
structions.
Chapter 9 presents in detail two solution techniques for the simple recourse prob
lem for the case of discrete random variables. They use the so-called 0- and A
representations of a piecewise linear function. It is also shown how the case of the
continuously distributed random variables can be treated.
Chapter 10 deals with the theorems concerning the convexity of probabilistic
constrained stochastic programming problems. Many of them are derived from the
results of Chapter 4.
Chapter 11 is devoted to programming under probabilistic constraints and the
methodologically similar problem: maximizing a probability under constraints. More
work has been done for the case of continuous random variables but at the end of
the chapter a few results are mentioned for the case of the discrete distribution. In
this chapter some nonlinear programming algorithms are presented which have been
tested for the solution of probabilistic constrained problems. I am sure there will be
others which will prove equally or more important in this respect in the future.
Chapter 12 describes the basic fact concerning the model: two-stage programming
under uncertainty, also called: stochastic programming with recourse. The most
important solution techniques were collected but many had to be left out.
Chapter 13 discusses the basic ideas about multi-stage stochastic programming
problems. This type of problems is very important and attracts great interest but,
in almost all cases we arrive at a very large problem; thus, only a few periods can
be taken into account, otherwise we cannot solve the problem. More new results are
urge~tly needed in this area to satisfy the demand of practice.
Preface xi
In Chapter 14 we present a selection of some special applied problems. Some of
them are network type problems and we may expect wide applicability for them in
the future.
Finally, Chapter 15 is devoted to distribution problems in stochastic programming.
This area is also rapidly developing, especially the analysis of stochastic combinatorial
optimization problems is under intensive investigation. A sample also from these
problems was taken.
Properties of the multivariate normal distribution are frequently used in the book,
therefore I felt it useful to include an appendix about it.
Except for Chapters 7, 13, and 14, all chapters are supplemented by Excersises
and Problems sections.
For about twenty years I have been teaching stochastic programming at the mas
ter's curriculum on operations research (as a specialised applied mathematical cur
riculum) at the Lorand Eotvos University of Budapest, Hungary and at the Ph.D.
program on operations research at Rutgers University of the United States of Amer
ica. For those who intend to teach it at a master's or Ph.D. level, I recommend
material for a one semester course. The materials for the master's and Ph.D. courses
can be the same but more proofs should be given to Ph.D. students.
Description of a one semester course material: 14 sessions, circa 3 hours each.
(1) Some basic statistical decision theoretic models, chosen from Chapter 7; formu
lation and solution of the newsboy problem. (2) Overview of the static stochastic
programming model constructions as described in Chapter 8. (3) Presentation of
the simple recourse problem and its reformulations for the case of discrete random
variables; the use of the 6- and A-representations and the main ideas of the relevant
solution techniques; the case of the uniform distribution. Use Chapter 9. (4) Basic
facts on logconcavity, Sections 4.1-4.4, 10.1-10.2. (5) Chapter 5 with emphasis on the
discrete moment problems. (6)-(7) Sections 6.1, 6.2, 6.5, 6.7, 6.8, and Chapter 1I.
(8)-(10) Chapter 12. (11) Chapter 13. (12) Sections 15.1-15.2, 15.5, and if we have
time, the main result of Section 15.6. (13)-(14) Applications from Chapter 14.
If we have a two semester course, then the above material can be presented in
more detailed form, more solution techniques can be included from Chapter 11, and
Chapter 13, Chapter 15 can almost entirely be included. In addition, the instructor
may require that the students should present at class articles from recent literature.
The book is the result of about 10 years of work. Similarly as my fellow stochastic
programmers feel when they finish a book written on the field, I am also not sure if
I solved the book-writing problem in a near optimal way. I am sure, however, that
the book will be useful for students and r~~earchers in operations research, statistics,
mathematics, and various fields of engineering, economics and other sciences.
I wish to express my thanks to RUTCOR, Rutgers University and partly the Air
Force! for supporting the writing of the book. My thanks should also be expressed to
those who read the manuscript and helped to correct the large number of errors: Ke
mal Giirsoy, Olga Fiedler, Lorant Porkolab and Jianmin Long of RUTCOR, Rutgers
IGra.nt numbers: AFORS-89-0512B, F49620-93-1-0041